FANDOM


$ \hat{\beta}_{OLS}=(X'X)^{-1}X'y $ By A1-Full Rank
$ E[\hat{\beta}_{OLS}|X]=E[(X'X)^{-1}X'y|X] $ By A2-Linearity
$ =E[(X'X)^{1}X'(X\beta+\epsilon)|X] =E[\beta+(X'X)^{1}X'\epsilon |X] \, $
$ =\beta + E[(X'X)^{1}X'\epsilon|X] \, $

Case 1: >A3Rsru Edit

$ E[\hat{\beta}_{OLS}|X]=\beta+0=\beta $ 

Under A3Rmi or stronger, the OLS estimator is unbiased.

Case 2: <=A3Rsru Edit

$ E[\hat{\beta}_{OLS}|X]=\beta+E[(X'X)^{1}X'\epsilon|X] $

Under A3Rsru or weaker, the OLS estimator is biased.

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